Home
Class 12
MATHS
Prove that [(a^2+b^2)/(a+b)]^(a+b)> a^a ...

Prove that `[(a^2+b^2)/(a+b)]^(a+b)> a^a b^b >{(a+b)/2}^(a+b)dot`

Text Solution

Verified by Experts

`([a+a+...a "times"]+[b+b+....b" times"])/(a+b)ge [a^ab^b]^((1)/(a+b))`
`ge (a+b)/(((1)/(a)+(1)/(a)+....a " times")+((1)/(b)+(1)/(b)+.....b" times"))`
or ` (a^2+b^2)/(a+b)ge[a^ab^b]^((1)/(a+b))ge (a+b)/(1+1)`
or ` ((a^2+b^2)/(a+b))^(a+b) ge a^a b^b ge ((a+b)/(2))^(a+b)`
or ` [(a^2+b^2)/(a+b)]^(a+b) gt a^a b^b gt {(a+b)/(2)}^(a+b)`
Promotional Banner

Topper's Solved these Questions

  • INEQUALITIES INVOLVING MEANS

    CENGAGE ENGLISH|Exercise Concept Application Eexercises 6.4|4 Videos

  • INEQUALITIES INVOLVING MEANS

    CENGAGE ENGLISH|Exercise EXERCISES (Single Correct answer type)|20 Videos

  • INEQUALITIES INVOLVING MEANS

    CENGAGE ENGLISH|Exercise Concept Application Eexercises 6.2|6 Videos

  • INEQUALITIES AND MODULUS

    CENGAGE ENGLISH|Exercise Single correct Answer|21 Videos

  • INTEGRALS

    CENGAGE ENGLISH|Exercise All Questions|764 Videos

Similar Questions

Explore conceptually related problems

If a ,b ,c are in G.P. then prove that (a^2+a b+b^2)/(b c+c a+a b)=(b+a)/(c+b)

Prove that b^2c^2+c^2a^2+a^2b^2> a b cxx(a+b+c)(a ,b ,c >0) .

Prove that (a b+x y)(a x+b y)>4a b x y(a , b ,x ,y >0)dot

Prove that (a b+x y)(a x+b y)>4a b x y(a , b ,x ,y >0)dot

If | vec a|= a\ a n d\ | vec b|=b, prove that ( vec a/(a^2)- vec b/(b^2))^2=(( vec a- vec b)/(a b))^2

If | vec a|=a and | vec b|=b , prove that ( vec a/(a^2)- vec b/(b^2))^2 = (( vec a- vec b)/(a b))^2 .

Prove that (b^2+c^2)/(b+c)+(c^2+a^2)/(c+a)+(a^2+b^2)/(a+b)> a+b+c

Prove that |(2ab,a^(2),b^(2)),(a^(2),b^(2),2ab),(b^(2),2ab,a^(2))|=-(a^(3)+b^(3))^(2) .

Prove that matrix [((b^(2)-a^(2))/(a^(2)+b^(2)),(-2ab)/(a^(2)+b^(2))),((-2ab)/(a^(2)+b^(2)),(a^(2)-b^(2))/(a^(2)+b^(2)))] is orthogonal.

Prove that: |(b+c)^2a^2a^2b^2(c+a)^2b^2c^2c^2(a+b)^2|=2a b c(a+b+c)^3